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Theorem rexuz2 11210
Description: Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
rexuz2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Distinct variable group:    n, M
Allowed substitution hint:    ph( n)

Proof of Theorem rexuz2
StepHypRef Expression
1 eluz2 11165 . . . . . 6  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n ) )
2 df-3an 987 . . . . . 6  |-  ( ( M  e.  ZZ  /\  n  e.  ZZ  /\  M  <_  n )  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
31, 2bitri 253 . . . . 5  |-  ( n  e.  ( ZZ>= `  M
)  <->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n ) )
43anbi1i 701 . . . 4  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) )
5 anass 655 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) ) )
6 anass 655 . . . . . 6  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
7 an12 806 . . . . . 6  |-  ( ( M  e.  ZZ  /\  ( n  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
86, 7bitri 253 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  ( M  <_  n  /\  ph ) )  <-> 
( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
95, 8bitri 253 . . . 4  |-  ( ( ( ( M  e.  ZZ  /\  n  e.  ZZ )  /\  M  <_  n )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
104, 9bitri 253 . . 3  |-  ( ( n  e.  ( ZZ>= `  M )  /\  ph ) 
<->  ( n  e.  ZZ  /\  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) ) )
1110rexbii2 2887 . 2  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) ) )
12 r19.42v 2945 . 2  |-  ( E. n  e.  ZZ  ( M  e.  ZZ  /\  ( M  <_  n  /\  ph ) )  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
1311, 12bitri 253 1  |-  ( E. n  e.  ( ZZ>= `  M ) ph  <->  ( M  e.  ZZ  /\  E. n  e.  ZZ  ( M  <_  n  /\  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    /\ w3a 985    e. wcel 1887   E.wrex 2738   class class class wbr 4402   ` cfv 5582    <_ cle 9676   ZZcz 10937   ZZ>=cuz 11159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-cnex 9595  ax-resscn 9596
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293  df-neg 9863  df-z 10938  df-uz 11160
This theorem is referenced by:  2rexuz  11211
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